task_type stringclasses 4
values | problem stringlengths 21 5.23k | answer stringlengths 1 8.29k | problem_tokens int64 11 1.16k | answer_tokens int64 1 2.04k |
|---|---|---|---|---|
math | ## 54. Math Puzzle 11/69
Two cans, filled with coffee, have the same shape and are made of the same material. The first can has a mass of $2 \mathrm{~kg}$ and is $12.0 \mathrm{~cm}$ high; the second has a mass of $1 \mathrm{~kg}$ and is $9.5 \mathrm{~cm}$ high.
How much coffee do the cans contain? | 1.92,0.95 | 99 | 9 |
math | 8.027. $\sin x \sin 3 x+\sin 4 x \sin 8 x=0$. | x_{1}=\frac{\pin}{7};x_{2}=\frac{\pik}{5},\quadn,k\inZ | 27 | 30 |
math | 833. Find all two-digit numbers that are equal to triple the product of their digits. | 15,24 | 20 | 5 |
math | 9.062. $\left(\frac{2}{5}\right)^{\log _{0.25}\left(x^{2}-5 x+8\right)} \leq 2.5$. | x\in[1;4] | 46 | 8 |
math | Example 5 A $98 \times 98$ chessboard is displayed on a computer screen, with its squares colored like a chessboard. You are allowed to select any rectangle with the mouse (the sides of the rectangle must lie on the grid lines), and then click the mouse button, which will change the color of each square in the rectangl... | 98 | 115 | 2 |
math | 1B. In the plane, two sets of parallel lines $p_{1}, p_{2}, \ldots, p_{13}$ and $q_{1}, q_{2}, \ldots, q_{7}$ are given such that the lines from the first set intersect with the lines from the second set. How many parallelograms are determined by these lines? | 1638 | 78 | 4 |
math | 1. Solve the inequality $\left(x^{2}+3 x+3\right)^{5 x^{3}-3 x^{2}} \leq\left(x^{2}+3 x+3\right)^{3 x^{3}+5 x}$. | x\in(-\infty;-2]\cup{-1}\cup[0;\frac{5}{2}] | 58 | 24 |
math | 15. Given the sequence $\left\{a_{n}\right\}(n \geqslant 0)$ satisfies $a_{0}=0$, for all non-negative integers $n$, we have
$$
a_{n+1}=2 \sqrt{30 a_{n}\left(a_{n}+1\right)}+11 a_{n}+5 .
$$
Find the general term formula for $a_{n}$. | a_{n}=-\frac{1}{2}+\frac{1}{4}(11+2 \sqrt{30})^{n}+\frac{1}{4}(11-2 \sqrt{30})^{n} | 97 | 52 |
math | 14. [40] Find an explicit, closed form formula for
$$
\sum_{k=1}^{n} \frac{k \cdot(-1)^{k} \cdot\binom{n}{k}}{n+k+1}
$$ | \frac{-1}{\binom{2n+1}{n}} | 55 | 16 |
math | 7. Let $p$ be a given odd prime, if the positive integer $k$ makes $\sqrt{k^{2}-p k}$ also a positive integer, then $k=$ | \frac{(p+1)^{2}}{4} | 38 | 13 |
math | 37. Given a segment and a line parallel to it. Using only a ruler, divide the segment in half. | M | 24 | 1 |
math | 5. [5 points] Given the numbers $\log _{\sqrt{2 x-8}}(x-4), \log _{(x-4)^{2}}(5 x-26), \log _{\sqrt{5 x-26}}(2 x-8)$. For which $x$ are two of these numbers equal, and the third one greater than them by 1? | 6 | 85 | 1 |
math | A gear wheel has 12 teeth on the smaller wheel and 32 teeth on the larger one. Due to a manufacturing defect, one tooth on the small wheel and one gap on the large wheel are faulty, causing them to rattle when they meet. How often will we hear the rattling if the time for one revolution of the large wheel is 3 seconds?... | 9 | 99 | 1 |
math | 7.271. $\left\{\begin{array}{l}3^{x} \cdot 2^{y}=972 \\ \log _{\sqrt{3}}(x-y)=2 .\end{array}\right.$ | (5;2) | 52 | 5 |
math | Find the minimum value of $\int_0^1 \{\sqrt{x}-(a+bx)\}^2dx$.
Please solve the problem without using partial differentiation for those who don't learn it.
1961 Waseda University entrance exam/Science and Technology | \frac{1}{450} | 57 | 9 |
math | Example 2 Solve the equation $5^{2 x}-6 \cdot 5^{x}+5=0$.
Translate the text above into English, keeping the original text's line breaks and format, and output the translation result directly.
Example 2 Solve the equation $5^{2 x}-6 \cdot 5^{x}+5=0$. | x_{1}=1,x_{2}=0 | 76 | 10 |
math | 16. Let $a, b, c$ be positive integers such that $a b+b c-c a=0$ and $a-c=101$. Find $b$.設 $a 、 b 、 c$ 爲正整數, 其中 $a b+b c-c a=0$ 而 $a-c=101$ 。求 $b$ 。 | 2550 | 86 | 4 |
math | 7.091. $\log _{2}\left(4 \cdot 3^{x}-6\right)-\log _{2}\left(9^{x}-6\right)=1$. | 1 | 44 | 1 |
math | 5. Given $a \neq b$, the remainder when polynomial $f(x)$ is divided by $x-a$ and $x-b$ are $c$ and $d$, respectively. Find the remainder when $f(x)$ is divided by $(x-a)(x-b)$. | \frac{-}{-b}x+\frac{-}{-b} | 59 | 15 |
math | Example 8. (I $150-17-3$ )
Find all polynomials in two variables that satisfy the following conditions:
1) $P$ is a homogeneous polynomial of degree $n$, i.e., for all real numbers $t$, $x$, $y$, we have
$$
P(t x, t y)=t^{\mathbb{n}} P(x, y) \text {; }
$$
2) For all real numbers $a, b, c$ we have
$$
\begin{array}{l}
P(... | P(x, y) = (x+y)^{n-1}(x-2y) | 163 | 20 |
math | Solve the following system of equations:
(1) $\quad x / y+y / x=6,6 \dot{5} 3846 \dot{1}$,
(2) $\quad 1 / x+1 / y=0,5 \dot{7} 6923 \dot{0}$.
(The numbers on the right side are repeating decimals. Read the solution to problem 1302, then convert the numbers on the right side to common fractions.) | x_1=13,y_1=2\quad\quadx_2=2,y_2=13 | 105 | 26 |
math | 306. The sum of the planar angles of a trihedral angle is $180^{\circ}$. Find the sum of the cosines of the dihedral angles of this trihedral angle. | 1 | 47 | 1 |
math | 10. Person A and Person B start from points $A$ and $B$ respectively (Person A starts from $A$), walking towards each other and continuously moving back and forth between the two points. Person A's speed is 4 times that of Person B. It is known that the distance between $A$ and $B$ is $S$ kilometers, where $S$ is a pos... | 105 | 285 | 3 |
math | 9. Cut a rectangular cardboard that is 90 cm long and 42 cm wide into small square pieces with integer centimeter sides and equal areas, without any remainder. The minimum number of pieces that can be cut is $\qquad$, and the total perimeter of all the square pieces cut in this way is $\qquad$ cm. | 105,2520 | 70 | 8 |
math | 356. Find the center and radius of the sphere
$$
x^{2}+y^{2}+z^{2}-2 x+6 y-6=0
$$ | C(1,-3,0),R=4 | 40 | 11 |
math | ## Task 1 - 220831
On a day in 1981, Cathrin asks her grandfather about his birth year. The grandfather, a friend of puzzle questions, replied:
"I am older than 65 years, but younger than 100 years. The year of my birth is not divisible by 2, 3, or 5. The remainder when this year is divided by 60 is not a prime numbe... | 1909 | 153 | 4 |
math | Example 2 Given that $x, y$ are both in the interval $(-2,2)$, and $x y=-1$, then what is the minimum value of the function $u=\frac{4}{4-x^{2}}+\frac{9}{9-y^{2}}$? | \frac{12}{5} | 62 | 8 |
math | Example 3 Let $P_{m}=\left\{A_{1}, A_{2}, \cdots, A_{m}\right\}$ be a partition of an $n$-element set $X$ (i.e., $A_{1}, A_{2}, \cdots, A_{m}$ is a partition of $X$). If one of the $A_{i}$ is further divided into two sets, this results in a partition $P_{m+1}$ of $X$, which consists of $m+1$ sets. $P_{m+1}$ is called a... | \frac{n!(n-1)!}{2^{n-1}} | 241 | 15 |
math |
2. We consider number sequences $a_{1}, a_{2}, a_{3}, \ldots$ such that $a_{n+1}=\frac{a_{n}+a_{1}}{a_{n}+1}$ holds for all $n \geqslant 1$.
(a) Suppose that $a_{1}=-3$. Compute $a_{2020}$.
(b) Suppose that $a_{1}=2$. Prove that $\frac{4}{3} \leqslant a_{n} \leqslant \frac{3}{2}$ holds for all $n \geqslant 2$.
| -3 | 145 | 2 |
math | Solve the following system of equations:
$$
\begin{gathered}
a^{3} x+a y+z=a^{2} \ldots \\
x+y+z=1 \ldots \\
8 x+2 y+z=4 \ldots
\end{gathered}
$$
What cases arise when $a=1, a=2, a=-3$? | \begin{aligned}&For\neq1,2,-3:\\&x=\frac{1}{+3},\quady=\frac{3a+2}{+3},\quadz=-\frac{2a}{+3}\\&For1:Indeterminatecase(\inftyinitelymanysolutions)\\&\text | 79 | 71 |
math | Given the circle $O$ and the line $e$, on which the point $A$ moves. The polar of point $A$ with respect to the circle intersects the perpendicular line to $e$ at point $A$ at point $M$. What is the geometric locus of point $M$? | y^{2}=r^{2}- | 62 | 8 |
math | 10.1. Try to find the smallest positive integer that cannot be expressed in the form $\frac{2^{a}-2^{b}}{2^{c}-2^{d}}$, where $a, b, c, d$ are all positive integers. | 11 | 55 | 2 |
math | 9.4. To the number $A$, consisting of eight non-zero digits, a seven-digit number, consisting of identical digits, was added, and the eight-digit number $B$ was obtained. It turned out that the number $B$ can be obtained from the number $A$ by rearranging some of the digits. What digit can the number $A$ start with if ... | 5 | 91 | 1 |
math | Positive sequences $\{a_n\},\{b_n\}$ satisfy:$a_1=b_1=1,b_n=a_nb_{n-1}-\frac{1}{4}(n\geq 2)$.
Find the minimum value of $4\sqrt{b_1b_2\cdots b_m}+\sum_{k=1}^m\frac{1}{a_1a_2\cdots a_k}$,where $m$ is a given positive integer. | 5 | 107 | 1 |
math | 62. Escalator. While on one of the escalators in the London Underground, I found that by walking 26 steps, I would reach the platform in $30 \mathrm{s}$. But if I walked 34 steps, the entire descent would take 18 s. How many steps are there on the escalator? Time is measured from the moment the top step begins to desce... | 46 | 95 | 2 |
math | Problem 8.8. In how many ways can all natural numbers from 1 to 200 be painted in red and blue so that the sum of any two different numbers of the same color is never equal to a power of two? | 256 | 50 | 3 |
math | 1. At what moment after noon do the hour and minute hands form a right angle for the first time? | \frac{180}{11} | 22 | 10 |
math | 8.041. $\cos 9 x-\cos 7 x+\cos 3 x-\cos x=0$. | x_{1}=\frac{\pik}{5};x_{2}=\frac{\pi}{6}(2+1),\quadk,\inZ | 27 | 33 |
math | The product of $10$ integers is $1024$. What is the greatest possible sum of these $10$ integers? | 1033 | 29 | 4 |
math | Question 8: Let $n \geq 2$ be a positive integer, and let real numbers $a_{1}, a_{2}, \ldots, a_{n}, b_{1}, b_{2}, \ldots, b_{n}$ satisfy: $\sum_{i=1}^{n} a_{i}=\sum_{i=1}^{n} b_{i}=1$. Find the smallest real number $\lambda$ such that $\lambda \cdot \Sigma_{1 \leq i<j \leq n}\left(a_{i} b_{j}-a_{j} b_{i}\right)^{2} \g... | n | 167 | 1 |
math | 3. $[\mathbf{3 0}]$ Let $x$ and $y$ be complex numbers such that $|x|=|y|=1$.
(a) [15] Determine the maximum value of $|1+x|+|1+y|-|1+x y|$.
(b) [15] Determine the maximum value of $|1+x|+|1+x y|+\left|1+x y^{2}\right|+\ldots+\left|1+x y^{2011}\right|-1006|1+y|$. | 2012\sqrt{2} | 121 | 9 |
math | 4. For a nonzero integer $a$, denote by $v_{2}(a)$ the largest nonnegative integer $k$ such that $2^{k} \mid a$. Given $n \in \mathbb{N}$, determine the largest possible cardinality of a subset $A$ of set $\left\{1,2,3, \ldots, 2^{n}\right\}$ with the following property:
$$
\text { for all } x, y \in A \text { with } ... | 2^{[\frac{n+1}{2}]} | 139 | 11 |
math | Find all positive integers $m, n$ such that $3^{m}-2^{n}=1$. | (1,1),(2,3) | 22 | 9 |
math | Which are the two consecutive odd numbers in the number sequence, the sum of whose squares is of the form $\frac{n(n+1)}{2}$, where $n$ is a natural number. | 1^{2}+3^{2}=\frac{4\cdot5}{2} | 41 | 19 |
math | ## Task 4 - 040924
Jutta, Günter, and Klaus are participating in the second stage of the Mathematics Olympiad.
(1) They work (not necessarily in this order) in rooms 48, 49, 50.
(2) Jutta and Günter are the same age, Klaus is one year older than Jutta.
(3) Their three mathematics teachers, Mr. Adler, Mr. Bär, and M... | Klaus,49 | 255 | 5 |
math | Example 5 For integer pairs $(a, b)(0<a<b<1000)$, a set $S \subseteq\{1,2, \cdots, 2003\}$ is called a "jump set" of the pair $(a, b)$: if for any element pair $\left(s_{1}, s_{2}\right)$, $s_{1} 、 s_{2} \in$ $S,\left|s_{1}-s_{2}\right| \notin\{a, b\}$.
Let $f(a, b)$ be the maximum number of elements in a jump set of ... | 668 | 157 | 3 |
math | ## Task 3 - 271223
a) For each natural number $n$, a function $f$ (with the domain of all real $x \neq 0$) is defined by
$$
f(x)=\sum_{k=0}^{n}(k-2) \cdot x^{k}
$$
Determine all natural numbers $n$ for which the function $f$ satisfies the equation $f(-1)=-f(1)$.
b) For each natural number $n$, a function $g$ (with ... | n=4n=5 | 192 | 6 |
math | 5. Given a grid board $2020 \times 2021$. Petya and Vasya are playing the following game. They take turns placing chips in free cells of the board. The player who, after their move, ensures that every $4 \times 4$ square contains a chip wins. Petya starts. Which player can guarantee a win regardless of the opponent's a... | Vasya | 86 | 3 |
math | G9.1 If $\sin \theta=\frac{3}{5}, a=\sqrt{\tan ^{2} \theta+1}$, find $a$. | \frac{5}{4} | 35 | 7 |
math | 6. Define the sequence $\left\{a_{n}\right\}: a_{n}$ is the last digit of $1+2+\cdots+n$, and $S_{n}$ is the sum of the first $n$ terms of the sequence $\left\{a_{n}\right\}$. Then $S_{2016}=$ $\qquad$ | 7066 | 79 | 4 |
math | 13.296. Two excavator operators must complete a certain job. After the first one worked for 15 hours, the second one starts and finishes the job in 10 hours. If, working separately, the first one completed $1 / 6$ of the job, and the second one completed $1 / 4$ of the job, it would take an additional 7 hours of their ... | 20 | 109 | 2 |
math | 6.5. There are nuts in three boxes. In the first box, there are six fewer nuts than in the other two boxes combined, and in the second box, there are ten fewer nuts than in the other two boxes combined. How many nuts are in the third box? Justify your answer. | 8 | 62 | 1 |
math | 9th Mexico 1995 Problem A1 N students are seated at desks in an m x n array, where m, n ≥ 3. Each student shakes hands with the students who are adjacent horizontally, vertically or diagonally. If there are 1020 handshakes, what is N? | 280 | 65 | 3 |
math | 66. After the tourists have walked 1 km and half of the remaining distance, they still have to walk a third of the entire distance and 1 km. What is the total distance? | 9 | 40 | 1 |
math | Consider the set $S$ of permutations of $1, 2, \dots, 2022$ such that for all numbers $k$ in the
permutation, the number of numbers less than $k$ that follow $k$ is even.
For example, for $n=4; S = \{[3,4,1,2]; [3,1,2,4]; [1,2,3,4]; [1,4,2,3]\}$
If $|S| = (a!)^b$ where $a, b \in \mathbb{N}$, then find the product $ab... | 2022 | 139 | 4 |
math | Solve in natural numbers $a,b,c$ the system \[\left\{ \begin{array}{l}a^3 -b^3 -c^3 = 3abc \\
a^2 = 2(a+b+c)\\
\end{array} \right.
\] | (4, 1, 3), (4, 2, 2), (4, 3, 1) | 63 | 28 |
math | E x a m p l e 1. Find the indefinite integral $\int(2 x+3)^{2} d x$. | \frac{1}{6}(2x+3)^{3}+C | 28 | 17 |
math | If we divide a number by 4, the remainder is 3, and if we divide it by 9, we get a remainder of 5. What is the remainder when we divide it by 36? | 23 | 45 | 2 |
math | Example 1.11. Find $\int \frac{\sin ^{3} x}{\cos ^{5} x} d x$. | \frac{1}{4\cos^{4}x}-\frac{1}{2\cos^{2}x}+C | 31 | 28 |
math | \section*{Exercise 2 - 121012}
In a right-angled Cartesian coordinate system, a parabola is given by the equation \(y=x^{2}\).
Provide an equation of a line that does not run parallel to the \(y\)-axis and intersects the parabola at exactly one point \(P\) with the abscissa 3! | 6x-9 | 80 | 4 |
math | 5. Let $P(x)$ be the polynomial of minimal degree such that $P(k)=720 k /\left(k^{2}-1\right)$ for $k \in\{2,3,4,5\}$. Find the value of $P(6)$. | 48 | 60 | 2 |
math | 3.186. $\frac{\sin ^{4} \alpha+\cos ^{4} \alpha-1}{\sin ^{6} \alpha+\cos ^{6} \alpha-1}=\frac{2}{3}$. | \frac{2}{3} | 54 | 7 |
math | Four different positive even numbers add up to 100. The third and fourth numbers are single-digit. If the first three numbers are multiplied by 12, 30, and 52 respectively, then the sum of these three resulting numbers is 2000. What are these four numbers? | 62,28,8,2or48,44,2,6 | 65 | 19 |
math | 1. (BUL) Find all the three-digit numbers for which one obtains, when dividing the number by 11, the sum of the squares of the digits of the initial number. | 550803 | 39 | 6 |
math | 3. In the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, $A B=B C=4, A A_{1}=2, P$ is a point on the plane $A_{1} B C$, and $\overrightarrow{D P} \cdot \overrightarrow{P B}=0$. Then, the area of the plane region enclosed by all points $P$ that satisfy the above conditions is $\qquad$. | \frac{36\pi}{5} | 103 | 10 |
math | 6. Find the number of matrices that satisfy two conditions:
3) the matrix has the form $\left(\begin{array}{lll}1 & * & * \\ * & 1 & * \\ * & * & 1\end{array}\right)$, where each * can take the value 0 or 1 and the rows of the matrix do not repeat. | 42 | 78 | 2 |
math | 1. (10 points). Two scientific and production enterprises supply the market with substrates for growing orchids. In the substrate "Orchid-1," pine bark is three times more than sand; peat is twice as much as sand. In the substrate "Orchid-2," bark is half as much as peat; sand is one and a half times more than peat. In... | 1:1 | 113 | 3 |
math | 6. (15 points) From a homogeneous straight rod, a piece of length $s=80 \mathrm{~cm}$ was cut. By how much did the center of gravity of the rod move as a result? | 40\mathrm{~} | 47 | 7 |
math | 267. Dubious Equality. The statement «$342=97$» can be made valid by inserting several algebraic signs between the digits, for example $(-3+4) \cdot 2=9-7$. Can the equality be given meaning without inserting any signs? | 342_{5}=97_{10} | 62 | 12 |
math | 3. Find any four pairwise distinct natural numbers $a, b, c, d$, for which the numbers $a^{2}+2 c d+ b^{2}$ and $c^{2}+2 a b+d^{2}$ are perfect squares. | 6=1\cdot6=2\cdot3 | 54 | 11 |
math | 4. A bag contains $m$ red balls and $n$ white balls, $m>n \geqslant 4$. Now, two balls are drawn at random. If the probability of drawing two balls of the same color is equal to the probability of drawing two balls of different colors, then the number of arrays $(m, n)$ that satisfy the relation $m+n \leqslant 40$ is $... | 3 | 92 | 1 |
math | One. (20 points) Given that $a$, $b$, and $c$ are three distinct real numbers, and the three quadratic equations
\[
\begin{array}{l}
x^{2} + a x + b = 0, \\
x^{2} + b x + c = 0, \\
x^{2} + c x + a = 0
\end{array}
\]
each pair of which has exactly one common root. Find the value of $a^{2} + b^{2} + c^{2}$. | 6 | 118 | 1 |
math | Each of $A, B, C$, and $D$ is a positive two-digit integer. These integers satisfy each of the equations
$$
\begin{aligned}
B & =3 C \\
D & =2 B-C \\
A & =B+D
\end{aligned}
$$
What is the largest possible value of $A+B+C+D$ ? | 204 | 77 | 3 |
math | 7. Find the positive integer $x$ that makes $x^{2}-60$ a perfect square
Find the positive integer $x$ that makes $x^{2}-60$ a perfect square | 16 \text{ or } 8 | 43 | 9 |
math | Example 8 Suppose $N$ consecutive positive integers satisfy the following conditions: the sum of the digits of the 1st number is divisible by 1, the sum of the digits of the 2nd number is divisible by 2, $\cdots$. The sum of the digits of the $N$th number is divisible by $N$. Find the maximum possible value of $N$.
| 21 | 81 | 2 |
math | 3. (5 points) If the product of 6 consecutive odd numbers is 135135, then the sum of these 6 numbers is $\qquad$ | 48 | 37 | 2 |
math | Express the number of diagonals of a convex n-gon. | \frac{\mathfrak{n}(\mathfrak{n}-3)}{2} | 13 | 19 |
math | Tyler rolls two $ 4025 $ sided fair dice with sides numbered $ 1, \dots , 4025 $. Given that the number on the first die is greater than or equal to the number on the second die, what is the probability that the number on the first die is less than or equal to $ 2012 $? | \frac{1006}{4025} | 78 | 13 |
math | Example 10 (45th IMO Problem) Find all real-coefficient polynomials $P(x)$ such that for all real numbers $a, b, c$ satisfying $a b+b c+c a=0$, we have
$$
P(a-b)+P(b-c)+P(c-a)=2 P(a+b+c) .
$$ | P(x)=\alphax^{4}+\betax^{2} | 71 | 15 |
math | 2. Find the sum of the first fifteen terms of an arithmetic progression if its third term is -5, and the fifth term is 2.4. | 202.5 | 32 | 5 |
math | We draw three cards from a Hungarian deck of cards.
a) If all three cards are of different suits, what is the probability that at least one of the three cards is an ace?
b) What is the probability of the latter, without any conditions?
c) What is the probability that we draw three cards of different suits, among whi... | \frac{169}{512},\frac{421}{1240},\frac{169}{1240} | 76 | 35 |
math | 14.7. a) In how many ways can a natural number $n$ be represented as the sum of $m$ non-negative integers, if the representations $n=x_{1}+\ldots+x_{m}$ and $n=y_{1}+\ldots+y_{m}$ are considered the same if and only if $x_{1}=y_{1}, \ldots, x_{m}=y_{m} ?$
b) The same question for representation as the sum of natural n... | C_{n+-1}^{n} | 106 | 9 |
math | 1. Let $[x]$ and $\{x\}$ be the integer and fractional parts of the number $x$. The integer part of the number $x$ is the greatest integer not exceeding $x$, and $\{x\}=x-[x]$. Find $x$ if $2 x+3[x]-5\{x\}=4$. | \frac{4}{3} | 74 | 7 |
math | 5. Two equal cones are arranged so that the axis of each is the generatrix of the other. The angles at the vertices in the axial sections of these cones are each $90^{\circ}$. Find the angle between the two generatrices along which these cones intersect. | 2\arccos\sqrt{\frac{2}{2+\sqrt{2}}} | 58 | 18 |
math | Problem 5.8. In the large family of Ivanovs, there are no twins. A reporter came to the Ivanovs to interview them.
During the interview, each of the children said: "I have an older brother." After thinking for a moment, the reporter was very surprised. But the head of the family explained that some of the children wer... | 8,10 | 111 | 4 |
math | 15th CanMO 1983 Problem 1 Find all solutions to n! = a! + b! + c! . | 3!=2!+2!+2! | 29 | 10 |
math | 16. (6 points) Two cars, A and B, start from two cities, A and B, which are 330 kilometers apart, and drive towards each other. Car A starts from city A, and after some time, car B starts from city B. The speed of car A is $\frac{5}{6}$ of the speed of car B. When the two cars meet, car A has traveled 30 kilometers mor... | 55 | 108 | 2 |
math | 9.077. $2^{x+2}-2^{x+3}-2^{x+4}>5^{x+1}-5^{x+2}$. | x\in(0;\infty) | 38 | 9 |
math | Find the angle between the lines $A C$ and $B D$, if the distance between the midpoints of segments $A D$ and $B C$ is equal to the distance between the midpoints of segments $A B$ and $C D$.
# | 90 | 55 | 2 |
math | 1. In a right-angled triangle with legs $a$ and $b, a>b$, the following equality holds:
$$
\lg \frac{a-b}{2}=\frac{1}{2}(\lg a+\lg b-\lg 2)
$$
Find the angles of the triangle. | \alpha=75,\beta=15 | 64 | 10 |
math | Example 8 There are 5 designated seats, and the 5 people sitting in these 5 seats do not know their designated numbers. When these 5 people randomly sit in the 5 seats, find:
(1) The probability that exactly 3 out of the 5 people sit in their designated seats;
(2) If the probability of these 5 people sitting in their d... | 2 | 107 | 1 |
math | Example 7. Solve the Bernoulli equation $y^{\prime}-x y=-x y^{3}$. | y^{2}(1+Ce^{-x^{2}})=1 | 25 | 14 |
math | 【Question 15】
In the plane, there are 50 lines, of which 20 are mutually parallel. These 50 lines can divide the plane into a maximum of $\qquad$ parts. | 1086 | 46 | 4 |
math |
Problem 2. Find all functions $f:(0, \infty) \rightarrow(0, \infty)$ such that
$$
f\left(f(x)+\frac{y+1}{f(y)}\right)=\frac{1}{f(y)}+x+1
$$
for all $x, y>0$.
(Dominik Burek)
| f(x)=x | 82 | 4 |
math | 20. (6 points) At 12 o'clock, the angle between the hour hand and the minute hand is 0 degrees. Thereafter, the moment when the hour hand and the minute hand form a 90-degree angle for the 6th time is $\qquad$ . (12-hour clock) | 3 | 67 | 1 |
math | Compute the number of positive integers less than or equal to $10000$ which are relatively prime to $2014$. | 4648 | 29 | 4 |
math | A school with 862 students will participate in a scavenger hunt, whose rules are:
I) The number of registrants must be a number between $\frac{2}{3}$ and $\frac{7}{9}$ of the total number of students in the school.
II) Since the students will be divided into groups of 11, the number of registrants must be a multiple ... | 8 | 100 | 1 |
math | . Let $\left(u_{n}\right)_{n \in \mathbb{N}^{*}}$ be the sequence such that $: \forall n \in \mathbb{N}^{*}, u_{n}=\frac{2 n+\cos (n)}{n \sin \left(\frac{1}{n}\right)+\sqrt{(n+1)(n+2)}}$.
Determine the limit of the sequence $\left(u_{n}\right)_{n \in \mathbb{N}^{*}}$. | 2 | 116 | 1 |
math | 2. Through the right focus of the hyperbola $x^{2}-\frac{y^{2}}{2}=1$, a line $l$ intersects the hyperbola at points $A$ and $B$. If a real number $\lambda$ makes $|A B|=\lambda$ such that there are exactly 3 lines $l$, then $\lambda=$
(Proposed by the Problem Committee) | 4 | 87 | 1 |
math | For each of the $9$ positive integers $n,2n,3n,\dots , 9n$ Alice take the first decimal digit (from the left) and writes it onto a blackboard. She selected $n$ so that among the nine digits on the blackboard there is the least possible number of different digits. What is this number of different digits equals to? | 4 | 78 | 3 |
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